** True or False** The following mathematical operation makes sense and is technically valid.
$$\nabla \cdot \nabla T(x,y,z)$$
1. Yes, it will produce a vector field.
2. Yes, it will produce a scalar field.
3. No, you can not take the divergence of a scalar field.
4. I don't remember what this means.
Note:
* Correct answer: B
A certain fluid has a velocity field given by $\mathbf{v} = x\hat{x} + z \hat{y}$. Which component(s) of the field contributed to "fluid flux" integral ($\int_S \mathbf{v}\cdot d\mathbf{A}$) through the x-z plane?
1. $v_x$
2. $v_y$
3. both
4. neither
Note:
* CORRECT ANSWER: B
* Only the vector perpendicular to the surface will contribute
* Fall 2016: 16 [78] 4 0 0
A certain fluid has a velocity field given by $\mathbf{v} = x\hat{x} + z \hat{y}$. If we intend to calculate the "fluid flux" integral ($\int_S \mathbf{v}\cdot d\mathbf{A}$) through the x-z plane, what is $d\mathbf{A}$ in this case? Be specific!
1. $\langle dx\,dy, 0, 0\rangle$
2. $\langle dx\,dz, 0, 0\rangle$
3. $\langle dy\,dz, 0, 0\rangle$
4. It's none of these
For the same fluid with velocity field given by $\mathbf{v} = x\hat{x} + z \hat{y}$. What is the value of the "fluid flux" integral ($\int_S \mathbf{v}\cdot d\mathbf{A}$) through the entire x-y plane?
1. It is zero
2. It is something finite
3. It is infinite
4. I can't tell without doing the integral
Note:
* CORRECT ANSWER: A
* The velocity field is parallel to the x-y plane every where and hence contributes no flux through the surface.
* Fall 2016: [89] 9 3 0 0
A rod (radius $R$) with a hole (radius $r$) drilled down its entire length $L$ has a mass density of $\frac{\rho_0\phi}{\phi_0}$ (where $\phi$ is the normal polar coordinate).
To find the total mass of this rod, which coordinate system should be used (take note that the mass density varies as a function of angle):
1. Cartesian ($x,y,z$)
2. Spherical ($r,\phi,\theta$)
3. Cylindrical ($s, \phi, z$)
4. It doesn't matter, just pick one.
Note:
* CORRECT ANSWER: C
* It makes the most sense from the geometry of the problem and writing the limits.
* Fall 2016: 0 0 [94] 6 0
Which of the following two fields has zero divergence?
| I | II |
|:-:|:-:|
| <img src ="./images/cq_left_field.png" align="center" style="width: 400px";/> | <img src ="./images/cq_right_field.png" align="center" style="width: 400px";/> |
1. Both do.
2. Only I is zero
3. Only II is zero
4. Neither is zero
5. ???
Note:
* CORRECT ANSWER: B
* Think about dE/dx and dE/dy
* Fall 2016: 7 [34] 13 43 3; (Asked them to consider dvx/dx and dvy/dy) 3 [90] 3 4 0
Which of the following two fields has zero curl?
| I | II |
|:-:|:-:|
| <img src ="./images/cq_left_field.png" align="center" style="width: 400px";/> | <img src ="./images/cq_right_field.png" align="center" style="width: 400px";/> |
1. Both do.
2. Only I is zero
3. Only II is zero
4. Neither is zero
5. ???
Note:
* CORRECT ANSWER: C
* Think about paddle wheel
* Fall 2016: 9 0 [89] 3 0